C₂×#9990 #232572

order	512
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#9990

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2686 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2686 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,16,20)(13,21,24,25)(14,17,15,19)(18,22,26,23);(7,10)(12,17)(13,22)(14,15)(19,20)(21,25)(23,24);(3,5)(4,7)(6,9)(8,10)(11,13,16,24)(12,18,20,26)(14,22,15,23)(17,21,19,25)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139768);